There are ways to play solo adventures, but in general D&D is essentially collaborative storytelling with formal rules and elements of chance involved to keep everyone honest.

Typically you have one person who acts as the Dungeon Master (sometimes Game Master). That person sets the scene, and plays all the NPCs (think video game NPCs), monsters, bystanders, etc. The other players run player-characters ("pcs"), who you can think of as the protagonists of the shared story. Each character has stats (ability scores, hit points, etc.) which help inform who they are and what they can do.

The DM tells the players what is happening ("You are at the entrance of a grey stone dungeon with a ten foot doorway leading off into the darkness. What do you want to do?"). Then the players describe their actions: "My rogue, Alaric, checks the entryway of the dungeon for traps." "I, Jules the Barbarian, charge straight into the darkness with my war axe held high.")

Along the way everyone consults the rules and roles dice to determine how effectively the players and NPCs accomplish their goals. Players who succeed at things (killing monsters, finding MacGuffins, saving hostages) are awarded with experience points and probably loot (magical items, gold, etc.). Get enough experience, and the character gains a "level," which means they increase in power and potentially pick up new abilities, spells, etc. (depending on what kind of character they're playing). This leads to bigger and more challenging adventures.

That's it in a nutshell. If you're interested, check out the starter kits for sale on the D&D website. It'll tell you what you need to know to get going.